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The Hilbert-Waring theorem

  • Melvyn B. Nathanson
Part of the Graduate Texts in Mathematics book series (GTM, volume 164)

Abstract

Waring’s problem for exponent k is to prove that the set of nonnegative integers is a basis of finite order, that is, to prove that every nonnegative integer can be written as the sum of a bounded number of kth powers. We denote by g(k) the smallest number s such that every nonnegative integer is the sum of exactly s kth powers of nonnegative integers. Waring’s problem is to show that g(k) is finite; Hilbert proved this in 1909. The goal of this chapter is to prove the Hilbert-Waring theorem: the kth powers are a basis of finite order for every positive integer k.

Keywords

Nonnegative Integer Hermite Polynomial Finite Order Polynomial Identity Fourth Power 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1996

Authors and Affiliations

  • Melvyn B. Nathanson
    • 1
  1. 1.Department of MathematicsLehman College of the City University of New YorkBronxUSA

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